Gr\" obner-Shirshov bases method for Novikov algebras

arXiv:1506.03466v1 [math.RA] 9 May 2015

¨ GROBNER-SHIRSHOV BASES METHOD FOR NOVIKOV ALGEBRAS∗ L. A. Bokut† School of Mathematical Sciences, South China Normal University Guangzhou 510631, P. R. China Sobolev Institute of Mathematics, Novosibirsk State University, Novosibirsk 630090, Russia [email protected]

Yuqun Chen‡ and Zerui Zhang School of Mathematical Sciences, South China Normal University Guangzhou 510631, P. R. China [email protected] [email protected]

Abstract: We establish Gröbner-Shirshov bases theory for Novikov algebras over a field of characteristic 0. As applications, a PBW type theorem in Shirshov form is given and we show that the word problem of Novikov algebras with finite homogeneous relations is solvable. We also construct a subalgebra of one generated free Novikov algebra which is ∗

Supported by the NNSF of China (11171118), the Research Fund for the Doctoral Program of Higher

Education of China (20114407110007) and the Program on International Cooperation and Innovation, Department of Education, Guangdong Province (2012gjhz0007). † Supported by RFBR 12-01-00329, LSS–3669.2010.1 and SB RAS Integration grant No.2009.97 (Russia) and Federal Target Grant Scientific and educational personnel of innovation Russia for 2009-2013 (government contract No.02.740.11.5191). ‡ Corresponding author.

1

not free. Key words: Keywords: Gröbner-Shirshov basis; Novikov algebra; commutative differential algebra; word problem. AMS 2010 Subject Classification: 17D25, 16S15, 13P10, 08A50

1

Introduction

Novikov algebras were introduced by I.M. Gel’fand, I.Ya. Dorfman [20], 1979 in connection with Hamiltonian operators in the formal calculus of variations and A.A. Balinskii, S.P. Novikov [4], 1985 in connection with linear Poisson brackets of hydrodynamic type. As it was pointed out in [26], 1985, E.I. Zelmanov answered to a Novikov’s question about simple finite dimensional Novikov algebras over a field of characteristic zero at the same year. He proved that there are no such algebras, see [45], 1987. In 1989, V.T. Filippov found first examples of simple infinite dimensional Novikov algebras of characteristic p ≥ 0 and simple finite dimensional Novikov algebras of characteristic p > 0, see [18]. J.M. Osborn [28, 29, 30], 1992-1994 gave the name Novikov algebra (he knew both papers [4, 20]) and began to classify simple finite dimensional Novikov algebras with prime characteristic p > 0 and infinite dimensional ones with characteristic 0, as well as irreducible modules [31], 1995, see also [32], 1995. Since that time there were quite a few papers on the structure theory (see, for example, X. Xu [41, 42, 43, 44], 1995-2000, C. Bai and D. Meng [1, 2, 3], 2001, L. Chen, Y. Niu and D. Meng [14], 2008, D. Burde and K. Dekimpe [11], 2006) and combinatorial theory of Novikov algebras, and irreducible modules over Novikov algebras, with applications to mathematics and mathematical physics. The present paper is on combinatorial method of Gröbner-Shirshov bases for Novikov algebras. Let us observe some combinatorial results. In [20], it was given an important observation by S.I. Gelfand that any differential commutative associative algebra is a Novikov algebra under the new product a ◦ b = (Da)b. This observation leads to a notion of the universal enveloping of a Novikov algebra (see and cf. [16]) that we use in the present paper. V.T. Filippov [19], 2001 proved that any Novikov nil-algebra of nil-index n with characteristic 0 is nilpotent (an analogy of Nagata-Higman theorem). He used essentially a Zelmanov theorem [46], 1988 that any Engel Lie algebra of index n with characteristic 0 is nilpotent. 2

By the way, Zelmanov [47], 1989 also proved the local nilpotency of any Engel Lie algebra of index n with any characteristic. In 2002, A. Dzhumadil’daev and C. Löfwall [16] found structure of a free Novikov algebra using trees and a free differential commutative algebra. We use essentially this results in here. Dzhumadil’daev [15], 2011 found another linear basis of a free Novikov algebra using Young tableaux. This result was essentially used by L. Makar-Limanov and U. Umirbaev [25] in a proof of Freiheitssatz theorem for Novikov algebras. Also they proved that the basic rank of the variety of Novikov algebras is one. In the present paper, we introduce Gröbner-Shirshov bases method for Novikov algebras, prove a PBW type theorem in Shirshov form for Novikov algebras, and prove algorithmically solvability of the word problem for Novikov algebras with finite number of homogeneous relations. Gröbner and Gröbner-Shirshov bases methods were invented by A.I. Shirshov, a student of A.G. Kurosh, for Lie algebras and implicitely for associative algebras [37], 1962 and nonassociative (commutative anti-commutative) algebras [36], 1954, by H. Hironaka for commutative topological algebras [21], 1964 and by B. Buchberger [10], 1965 for commutative algebras. As a prehistory, see A.I. Zhukov [48], 1950, another Kurosh’s student. Gröbner-Shirshov (Gröbner) bases methods deal with varieties and categories of (differential, integro-differential, PBW, Levitt, Temperley-Lieb, Iwahory-Hecky, quadratic, free products of two, over a commutative algebra, . . . ) associative algebras, (plactic, Chines, inverse, . . . ) semigroup algebras, (Coxeter, braid, Artin-Tits, Novikov-Boon, . . . ) group algebras, semiring algebras, Lie (restricted, super-, semisimple, Kac-Moody, quantum, Drinfeld-Kohno, over a commutative algebra, metabelian, . . . ) algebras, associative conformal algebras, Loday’s (Leibniz, di-, dendriform) algebras, Rota-Baxter algebras, pre-Lie (i.e., right symmetric) algebras, (simplicial, strict monoidal, . . . ) categories, nonassociative (commutative, anti-commutative, Akivis, Sabinin, . . . ) algebras, symmetric (nonsymmetric) operads, Ω-algebras, modules, and so on. Gröbner-Shirshov bases method is useful in homological algebra (Anick resolutions, (Hochschild) cohomology rings of (Levitt, plactic, . . . ) algebras), in proofs of PBW type theorems (Lie algebra – associative algebra, Lie algebra – pre-Lie algebra, Leibniz algebra – associative dialgebra, Akivis algebra – nonassociative algebra, Sabinin algebra - modules), in algorithmic problems of algebra (solvable and unsolvable algorithmic problems), in the theory of au3

tomatic groups and semigroups, in independent constructions of Hall, Hall-Shirshov and Lyndon-Shirshov bases of a free Lie algebra, on embedding theorems and many other applications. For details one may see, for example, new surveys [6] and [34]. In this paper we prove a PBW type theorem in Shirshov form for Novikov algebras. The first of this kind of theorems is the following (see [7, 8, 9]). PBW theorem in Shirshov form. Let L = Lie(X|S) be a Lie algebra, presented by generators X and defining relations S over a field k , U(L) = khX|S (−) i the universal enveloping associative algebra of L (here S ⇒ S (−) using [x, y] ⇒ xy−yx). Then S is a Lie Gröbner-Shirshov basis in Lie(X) if and only if S (−) is an associative Gröbner-Shirshov basis in khXi. Corollary. Let S be a Lie Gröbner-Shirshov basis (in particular, S be a multiplication table of L). Then (i) A linear basis of U(L) consists of words u1 u2 . . . uk , k ≥ 0, where ui ’s are S (−) irreducible Lyndon-Shirshov words in X, u1 ≤ u2 ≤ · · · ≤ uk in lexicographical order (meaning a > ab, if b 6= 1) (in particular, a linear basis of U(L) is PBW one if S is a miltiplication table of L). (ii) A linear basis of U(L) consists of words [u1 ][u2 ] . . . [uk ], k ≥ 0, where [ui ]’s are S-irreducible Lyndon-Shirshov Lie words in X, u1 ≤ u2 ≤ ... ≤ uk in lexicographical order. (iii) A linear basis of L consists of words [u], where [u]’s are S-irreducible LyndonShirshov Lie words in X. For Novikov algebras we prove the following PBW type theorem in Shirshov form. Let Nov(X) be a free Novikov algebra, k{X} be a free commutative differential algebra, S ⊆ Nov(X) and S c a Gröbner-Shirshov basis in k{X}, which is obtained from S by Buchberger-Shirshov algorithm in k{X}. Then (i) S ′ = {uD m s | s ∈ S c , u ∈ [D ω X], wt(uD ms) = −1, m ∈ N} is a Gröbner-Shirshov basis in Nov(X). (ii) The set Irr(S ′ ) = {w ∈ [D ω X] | w 6= uD t s, u ∈ [D ω X], t ∈ N, s ∈ S c , wt(w) = −1} = Nov(X) ∩ Irr[S c ] is a linear basis of Nov(X|S). Thus, any Novikov algebra Nov(X|S) is embeddable into its universal enveloping commutative differential algebra k{X|S}. 4

Using Buchberger-Shirshov algorithm, we prove the algorithmic solvability of the word problem for commutative differential algebras with finite number of D ∪ X-homogeneous defining relations and the algorithmic solvability of the word problem for Novikov algebras with finite number of X-homogeneous defining relations. For Lie algebras it was proved by Shirshov in his original paper [37], see also [38]. In general, word problem for Lie algebras is unsolvable, see [5]. For Novikov algebras it remains unknown. So far, the word problem (membership problem) for commutative differential algebras is solved mainly for the following cases [23]: radical ideals, isobaric (i.e., homogeneous with respect to derivation operators) ideals, ideals with a finite or parametrical standard basis, and ideals generated by a composition of two differential polynomials (under some additional assumptions). At last, we prove that the variety of Novikov algebras is not a Schreier one, i.e., not each subalgebra of a free Novikov algebra is free. The most famous Schreier variety are the variety of groups [33], the variety of non associative algebras [24], the variety of (non associative) commutative and anti-commutative algebras [36], the variety of Lie algebras [35, 40]. For more details, see [13, 39].

2

Free Novikov Algebra

A non-associative algebra A = (A, ◦) is called a right-Novikov algebra [15], if it satisfies the identities x ◦ (y ◦ z) − (x ◦ y) ◦ z = x ◦ (z ◦ y) − (x ◦ z) ◦ y, x ◦ (y ◦ z) = y ◦ (x ◦ z). A is called a left-Novikov algebra if it satisfies x ◦ (y ◦ z) − (x ◦ y) ◦ z = y ◦ (x ◦ z) − (y ◦ x) ◦ z, (x ◦ y) ◦ z = (x ◦ z) ◦ y. In this paper, we only consider right-Novikov algebras. In the papers [15, 16], the authors constructed the free Novikov algebra N(X) generated by X as follows: A Young diagram is a set of boxes with non-increasing numbers of 5

boxes in each row. Rows and columns are numbered from top to bottom and from left to right. Let p be the number of rows and ri be the number of boxes in the ith row. To construct Novikov diagram, we need to complement Young diagram by one box in the first row. To construct Novikov tableaux on a well-ordered set X, we need to fill Novikov diagrams by elements of X. Denote by ai,j an element of X in the box that is the cross of the ith row by the jth column. The filling rule is the following: (a) ai,1 ≥ ai+1,1 , if ri = ri+1 , i = 1, 2, . . . , p − 1; (b) The sequence ap,2 · · · ap,rp ap−1,2 · · · ap−1,rp−1 · · · a1,2 · · · a1,r1 +1 is nondecreasing. Correspond to such a Novikov tableau an element w = Yp ◦ (Yp−1 ◦ (· · · ◦ (Y2 ◦ Y1 ) · · · )) (right-normed bracketing), where Yi = (· · · ((ai,1 ◦ ai,2 ) ◦ ai,3 ) · · · ◦ ai−1,ri −1 ) ◦ ai,ri , 2 ≤ i ≤ p, Y1 = (· · · ((a1,1 ◦ a1,2 ) ◦ a1,3 ) · · · ◦ a1,r1 ) ◦ a1,r1 +1 (each Yj left-normed bracketing). In this case, we say w has degree rp + rp−1 + · · · + r1 + 1. We call such a w as a Novikov tableau as well. Such elements consist of a linear basis of a free Novikov algebra generated by X and we denote such free Novikov algebra as N(X), see [15, 16]. A commutative differential algebra A = (A, ·, D) is a commutative associative algebra with one linear operator D : A → A such that for any a, b ∈ A, D(ab) = (Da)b + a(Db). We call such a D a derivation of A. Given a well-ordered set X = {a, b, c, . . . }, denote D ω X = {D i a | i ∈ N, a ∈ X}, [D ω X] the free commutative monoid generated by D ω X and k a field of characteristic 0. Let D(1) = 0, D 0a = a, D(D i a) = D i+1 a, D(αu + βv) = αDu + βDv and D(uv) = (Du) · v + u · D(v) for any a ∈ X, α, β ∈ k, u, v ∈ [D ω X] (· is often omitted). Then (k[D ω X], ·, D) is a free commutative differential algebra over k, see [22]. From now on we denote a as a[−1], D i+1 a as a[i], and (k[D ω X], ·, D) as k[D ω X] or k{X}. Then k{X} has a k-basis as the set (also denote as [D ω X]) of all words of the form w = an [in ]an−1 [in−1 ] · · · a1 [i1 ] or w = 1, where at ∈ X, it ≥ −1, 1 ≤ t ≤ n, n ∈ N and (in , an ) ≥ (in−1 , an−1 ) ≥ · · · ≥ (i1 , a1 ) lexicographically. For such w 6= 1, we define the weight of w, denoted by wt(w), to 6

be wt(w) = i1 +i2 +· · ·+in ; the length of w, denoted by |w|, to be |w| = n; and the D ∪Xlength of w, denoted by |w|D∪X , to be |w|D∪X = wt(w) + 2n, which is exactly the number of D and generators from X that occur in w. For w = 1, define wt(w) = |w| = |w|D∪X = 0. Furthermore, if we define ◦ as f ◦ g = (Df )g,

f, g ∈ k{X},

then (k{X}, ◦) becomes a right-Novikov algebra. Its subspace, Nov(X) , spank {w ∈ [D ω X] | wt(w) = −1} is a subalgebra of (k{X}, ◦) (as Novikov algebra). In [16], the authors showed that the Novikov algebra homomorphism ϕ : N(X) −→ Nov(X), induced by ϕ(a) = a[−1], is an isomorphism. Therefore, Nov(X) is a free Novikov algebra generated by X, which has a k-basis {w ∈ [D ω X] | wt(w) = −1}.

3

Composition-Diamond Lemmas

3.1

Monomial order

We order [D ω X] as follows. For any a[i], b[j] ∈ D ω X, define a[i] < b[j] ⇔ (i, a) < (j, b) lexicographically. For any w = an [in ] · · · a1 [i1 ] ∈ [D ω X] with an [in ] ≥ · · · ≥ a1 [i1 ], define ord(w) , (|w|, an[in ], . . . , a1 [i1 ]). Then, for any u, v ∈ [D ω X] we define u < v ⇔ ord(u) < ord(v) lexicographically. It is clear that this is a well order on [D ω X]. We will use this order throughout this paper. For any f ∈ k{X}, f means the leading word of f . We denote the coefficient of f as LC(f ). 7

Lemma 3.1. Let the order < on [D ω X] be as above. Then (i) u < v ⇒ u · w < v · w, Du < Dv for any u, v, w ∈ [D ω X], u 6= 1. (ii) u < v ⇒ w ◦ u < w ◦ v, u ◦ w < v ◦ w for any u, v, w ∈ [D ω X] \ {1}. Proof. (i) Noting that · is commutative and associative, it is easy to see that u < v ⇒ u · w < v · w. For any w = an [in ] · · · a1 [i1 ] 6= 1, with an [in ] ≥ · · · ≥ a1 [i1 ], we have ord(Dw) = (|w|, an [in + 1], . . . , a1 [i1 ]), so u < v ⇒ Du < Dv. (ii) For any u, v, w ∈ [D ω X] \ {1}, we have u ◦ w = (Du)w = Du · w and v ◦ w = (Dv)w = Dv · w, so u < v ⇒ u ◦ w < v ◦ w. By the same reasoning, u < v ⇒ w ◦ u < w ◦ v.

3.2

S-words

For any S ⊆ k{X}, we denote Id[S] the ideal of k{X} generated by S and k{X|S} , k{X}/Id[S] the commutative differential algebra generated by X with defining relations S. Since Id[S] is closed under · and the derivative operator D, we have Id[S] = spank {uD t s | u ∈ [D ω X], t ∈ N, s ∈ S}. For any u ∈ [D ω X], t ∈ N, s ∈ S, we call uD ts an S-word in k{X}. We call uD t s an S-word in Nov(X) if wt(uD ts) = −1 and S ⊆ Nov(X). Suppose S ⊆ Nov(X) and denote Id(S) the ideal of Nov(X) generated by S. Then we have the following lemma. Lemma 3.2. Suppose S ⊆ Nov(X). Then Id(S) = spank {uD t s | u ∈ [D ω X], t ∈ N, s ∈ S, wt(uD t s) = −1}. Proof. It is clear that the right part is an ideal that contains S. We just need to show that uD t s ∈ Id(S) whenever wt(uD ts) = −1. Since wt(uD ts) = −1, we have uD t s = c1 [i1 ] · · · cn [in ]a1 [−1] · · · am [−1](D t s)b1 [−1] · · · bt [−1], where u = c1 [i1 ] · · · cn [in ]a1 [−1] · · · am [−1]b1 [−1] · · · bt [−1], m = i1 + · · · + in and in ≥ in−1 ≥ · · · ≥ i1 ≥ 0. So the lemma will be clear if we show 8

(i) (D t s)b1 [−1] · · · bt [−1] ∈ Id(S) whenever s ∈ S; (ii) c[p]a1 [−1] · · · ap [−1]f ∈ Id(S) whenever f ∈ Id(S). To prove (i), we use induction on t. If t = 0, it is clear. Suppose that it holds for all t ≤ n. Then (D n+1 s)b1 [−1] · · · bn+1 [−1] = ((D n s)b1 [−1] · · · bn [−1]) ◦ bn+1 [−1] X − (D n s)b1 [−1] · · · (Dbi [−1]) · · · bn [−1] · bn+1 [−1] 1≤i≤n n

= ((D s)b1 [−1] · · · bn [−1]) ◦ bn+1 [−1] X − bi [−1] ◦ ((D n s)b1 [−1] · · · bi−1 [−1]bi+1 [−1] · · · bn+1 [−1]) 1≤i≤n

∈ Id(S). To prove (ii), we use induction on p. If p = 0, it is clear. Suppose that it holds for all p ≤ n. Then c[n + 1]a1 [−1] · · · an+1 [−1]f = (c[n]a1 [−1] · · · an+1 [−1]) ◦ f X − c[n]a1 [−1] · · · (Dai [−1]) · · · an+1 [−1] · f 1≤i≤n+1

= (c[n]a1 [−1] · · · an+1 [−1]) ◦ f X − c[n]a1 [−1] · · · ai−1 [−1]ai+1 [−1] · · · an+1 [−1] · (ai [−1] ◦ f ) 1≤i≤n+1

∈ Id(S). So Id(S) = spank {uD t s | u ∈ [D ω X], t ∈ N, s ∈ S, wt(uD t s) = −1}. Let S be a subset of k{X}. We call S homogeneous (weight homogeneous, D ∪ XP homogeneous, resp.), if for any f = qj=1 βj wj ∈ S, we have |w1 | = · · · = |wq | (wt(w1) =

· · · = wt(wq ), |w1|D∪X = · · · = |wq |D∪X , resp.). We have the following lemma immediately. Lemma 3.3. Let S ⊆ k{X}, f =

P

i∈I

βi ui D ti si , where each βi ∈ k, ui ∈ [D ω X], si ∈

S, ti ∈ N. If f and S are homogeneous (weight homogeneous, D ∪ X-homogeneous, 9

resp.), then we can suppose that |ui D ti si | = |f | (wt(ui D ti si ) = wt(f ), |ui D ti si |D∪X = |f |D∪X , resp.) for any i ∈ I.

3.3

Composition-Diamond lemma for commutative differential algebras

The idea of this subsection is essentially the same as the construction of standard differential Gröbner basis in [17, 27], in which the authors deal with more general case with several derivative operators. For any u, v ∈ [D ω X], we always denote lcm(u, v) the least common multiple of u, v in [D ω X]. We call lcm(u, v) a non-trivial least common multiple of u and v if |lcm(u, v)| < |uv|. For any f, g ∈ S ⊆ k{X}, if w = lcm(D t1 f , D t2 g) is a non-trivial least common multiple of D t1 f and D t2 g, then we call [D t1 f, D t2 g]w =

1 1 w| t1 w| − t α1 D f 7→D 1 f α2 Dt2 g7→Dt2 g

a composition for D t1 f ∧D t2 g corresponding to w, where α1 = LC(D t1 f ), α2 = LC(D t2 g). P For a polynomial h ∈ k{X}, we say h ≡ 0 mod(S, w) if h = γi ui D ti si , where each γi ∈ k, ui D ti si is an S-word and ui D ti si < w. Denote h ≡ h′ mod(S, w) if h −

h′ ≡ 0 mod(S, w). The composition [D t1 f, D t2 g]w is trivial mod(S, w) if [D t1 f, D t2 g]w ≡ 0 mod(S, w). For f, g ∈ S, w = lcm(D t1 f , D t2 g), if w = f or w = g, then the composition is called inclusion; Otherwise, the composition is called intersection. Definition 3.1. Let S be a non-empty subset of k{X}. Then the set S is called a GröbnerShirshov basis in k{X} if all compositions of S in k{X} are trivial. Theorem 1. (Composition-Diamond lemma for commutative differential algebras) [17, 27] Let < be the monomial order on k{X} as before and S a non-empty subset of k{X}. Let Id[S] be the ideal of k{X} generated by S. Then the following statements are equivalent. (i) S is a Gröbner-Shirshov basis in k{X}. 10

(ii) 0 6= h ∈ Id[S] ⇒ h = uD t s for some s ∈ S, u ∈ [D ω X], t ∈ N. (iii) Irr[S] = {w ∈ [D ω X] | w 6= uD t s, u ∈ [D ω X], t ∈ N, s ∈ S} is a linear basis for k{X|S}.

Buchberger-Shirshov algorithm: If a subset S ⊂ k{X} is not a Gröbner-Shirshov basis then one can add all non-trivial compositions of S to S. Continuing this process repeatedly, we finally obtain a Gröbner-Shirshov basis S c that contains S. Such a process is called Buchberger-Shirshov algorithm.

3.4

Composition-Diamond lemma for Novikov algebras

For any u, v, w ∈ [D ω X], we call w a common multiple of u and v in Nov(X) if wt(w) = −1 and w is a common multiple of u and v in [D ω X]; w is a non-trivial common multiple of u and v in Nov(X) if w is a common multiple of u and v in Nov(X) such that w 6= uvw ′ for any w ′ ∈ [D ω X]. Let f, g ∈ Nov(X) and w a non-trivial common multiple of D t1 f and D t2 g in Nov(X). Then a composition of D t1 f ∧ D t2 g relative to w is defined as (D t1 f, D t2 g)w =

1 1 w| t − w| t , t1 f 1 D f → 7 D α1 α2 D 2 g7→Dt2 g

where α1 = LC(D t1 f ) and α2 = LC(D t2 g). Suppose that S ⊆ Nov(X) and h ∈ Nov(X). Then we say h ≡ 0 mod(S, w) if P h= βi ui D ti si , where each βi ∈ k, ui D ti si is an S-word such that wt(ui D ti si ) = −1 and ui D ti si < w. The composition (D t1 f, D t2 g)w is trivial mod(S, w) if (D t1 f, D t2 g)w ≡

0 mod(S, w). Let w = lcm(D t1 f , D t2 g)d1 [m1 ] · · · dp [mp ]c1 [−1] · · · cq [−1] be a non-trivial common multiple of D t1 f and D t2 g in Nov(X), where m1 ≥ · · · ≥ mp > 0. Then we say w to be critical if one of the following holds: (i) If wt(lcm(D t1 f , D t2 g) > −1, then d1 [m1 ] · · · dp [mp ] is empty. (ii) If wt(lcm(D t1 f , D t2 g) = −1, then d1 [m1 ] · · · dp [mp ]c1 [−1] · · · cq [−1] is empty. 11

(iii) If wt(lcm(D t1 f , D t2 g)) < −1, then wt(lcm(D t1 f , D t2 g)d1 [m1 ] · · · dp−1[mp−1 ]) < −1 and wt(lcm(D t1 f , D t2 g)d1 [m1 ] · · · dp [mp ]) ≥ −1. Definition 3.2. Let S be a non-empty subset of Nov(X). Then the set S is called a Gröbner-Shirshov basis in Nov(X) if all compositions of S in Nov(X) are trivial. Lemma 3.4. Suppose that the compositions (D t1 f, D t2 g)w are trivial for all critical common multiples w of D t1 f and D t2 g, where f, g ∈ S, t1 , t2 ∈ N. Then S is a GröbnerShirshov basis in Nov(X). Proof. Noting that any common multiple of D t1 f and D t2 g contains some critical common multiple w of D t1 f and D t2 g, the result follows. Lemma 3.5. Suppose that S is a Gröbner-Shirshov basis in Nov(X), f, g ∈ S and w = uD t f = vD t′ g ∈ [D ω X], wt(w) = −1. Then

1 uD t f α1

−

′ 1 vD t g α2

≡ 0 mod(S, w),

t′

where α1 = LC(uD t f ) and α2 = LC(vD g). Proof. If u = u′D t′ g for some u′ ∈ [D ω X], then v = u′ D t f . Thus 1 1 ′ uD t f − vD t g α1 α2 1 ′ t 1 1 ′ t 1 ′ t′ ′ ′ ′ u (D g)D t f − u (D f )D t g + u (D f )D t g − u′ (D t f )D t g = α1 α1 α2 α1 α2 α2 1 1 1 ′ 1 ′ = (D t′ g − D t g)u′D t f − (D t f − D t f )u′D t g α1 α2 α2 α1 ≡ 0 mod(S, w). Otherwise, w a non-trivial common multiple of D t f and D t′ g in Nov(X). Since S is a Gröbner-Shirshov basis, by definition we have

1 uD t f α1

−

′ 1 vD t g α2

≡ 0 mod(S, w).

Lemma 3.6. Let S be a non-empty subset of Nov(X). Denote Irr(S) = {w ∈ [D ω X] | w 6= uD ts, u ∈ [D ω X], t ∈ N, s ∈ S, wt(w) = −1}. Then for all h ∈ Nov(X), we have X h=

βi ui D ti si +

ui D ti si ≤h

X

γ j wj ,

wj ≤h

where each βi , γj ∈ k, ui ∈ [D ω X], ti ∈ N, si ∈ S, wj ∈ Irr(S), and wt(ui D ti si ) = wt(wj ) = −1. 12

Proof. By induction on h, we have the result. Theorem 2. (Composition-Diamond lemma for Novikov algebras) Let S be a non-empty subset of Nov(X) and Id(S) be the ideal of Nov(X) generated by S. Then the following statements are equivalent. (i) S is a Gröbner-Shirshov basis in Nov(X). (ii) 0 6= h ∈ Id(S) ⇒ h = uD t s for some s ∈ S, u ∈ [D ω X], t ∈ N. (iii) Irr(S) = {w ∈ [D ω X] | w 6= uD t s, u ∈ [D ω X], t ∈ N, s ∈ S, wt(w) = −1} is a linear basis for Nov(X|S) , Nov(X)/Id(S). Proof. (i) ⇒ (ii). Let S be a Gröbner-Shirshov basis and 0 6= h ∈ Id(S). Then h has P an expression h = ni=1 βi ui D ti si , where each 0 6= βi ∈ k, ui ∈ [D ω X], ti ∈ N, si ∈

S, wt(ui D ti si ) = −1. Denote wi = ui D ti si , i = 1, 2, . . . , n. We may assume without loss of generality that w1 = w2 = · · · = wl > wl+1 ≥ wl+2 ≥ . . .

for some l ≥ 1. Then w1 ≥ h. We show the result by induction on (w1 , l), where for any l, l′ ∈ N and w, w ′ ∈ [D ω X], (w, l) < (w ′ , l′ ) means lexicographically less. We call (w1 , l) the height of h. If h = w1 , then the result is obvious. Now suppose that w1 > h. Then l > 1 and u1 D t1 s1 = u2 D t2 s2 . By Lemma 3.5, we have β1 u1 D t1 s1 + β2 u2 D t2 s2 α1 α1 β1 + α2 β2 = β1 (u1 D t1 s1 − u2 D t2 s2 ) + u2 D t2 s2 α2 α2 α1 β1 + α2 β2 u2 D t2 s2 mod(S, w1 ), ≡ α2 where αi = LC(ui D ti si ), i = 1, 2. Thus, n

h =

m

X X α1 β1 + α2 β2 ′ ′ u2 D t2 s2 + βi ui D ti si + γj vj D tj s′j , vj D tj s′j < w1 , α2 i=3 j=1

which has height < (w1 , l). Now the result follows by induction hypothesis. 13

(ii) ⇒ (iii). By Lemma 3.6, the set Irr(S) generates the algebra Nov(X|S) as a P k-vector space. On the other hand, suppose that 1≤i≤n γi wi = 0 in Nov(X|S), where P each 0 6= γi ∈ k, wi ∈ Irr(S) and w1 > w2 > · · · > wn . Then we have 1≤i≤n γi wi = P tj / Irr(S), which contradicts to 1≤j≤m βj uj D sj 6= 0 in Nov(X). So by (ii) we get w1 ∈ the choice of w1 .

(iii) ⇒ (i). For any f, g ∈ S, t1 , t2 ∈ N, denote w a non-trivial common multiple of D t1 f and D t2 g. Then by Lemma 3.6, we have X X βi ui D ti si + γ j wj , (D t1 f, D t2 g)w = ui D ti si . . . . Then i∈I αi ϕ(wi ) = 0 in Nov(X|ϕ(S)) and ϕ(w1 ) = uD t S for some u ∈ [D ω X], s ∈ S ′ , t ∈ N by Theorem 2, which contradicts to the choice of w1 .

4

Applications

4.1

An example

In the paper [12], the authors list a lot of left-Novikov algebras in low dimensions. We can get their corresponding right-Novikov algebras using a ◦op b , b ◦ a, see also [15]. Example 4.1. ([12]) Let X = {e1 , e2 , e3 , e4 }, S = {e2 [0]e1 [−1] = e3 [−1], e3 [0]e1 [−1] = e4 [−1], ei [0]ej [−1] = 0, if (i, j) ∈ / {(2, 1), (3, 1)}, 1 ≤ i, j ≤ 4}. Then S is a GröbnerShirshov basis in Nov(X). It follows from Theorem 2 that {e1 [−1], e2 [−1], e3 [−1], e4 [−1]} is a linear basis of the Novikov algebra Nov(X|S). Proof. Denote fij :

ei [0]ej [−1] =

X

l αij el [−1] ∈ S, 1 ≤ i, j ≤ 4.

1≤l≤4

Before checking the compositions, we prove the following claims. Claim (i): Let w = ei [n]ei1 [−1] · · · ein+1 [−1], n ≥ 0. Then w = each uj D tj sj ≤ w, sj ∈ S if il 6= 1 for some 1 ≤ l ≤ n + 1.

P

αj uj D tj sj , where

We show Claim (i) by induction on n. For n = 0 or 1, the result follows immediately.

15

Suppose t ≥ 2 and the result holds for any n < t. Then w = ei [t]ei1 [−1] · · · eit+1 [−1] X m = D t (ei [0]ei1 [−1] − αi,i e [−1])ei2 [−1] · · · eit+1 [−1] 1 m 1≤m≤4

X

−

Ctp ei [p]ei1 [t

− 1 − p]ei2 [−1] · · · eit+1 [−1]

0≤p≤t−1

+

X

m αi,i e [t − 1]ei2 [−1] · · · eit+1 [−1]. 1 m

1≤m≤4

If for any 1 ≤ l ≤ n + 1, il 6= 1, then by induction hypothesis, the result follows immediately. Otherwise, say i1 = 1, then il 6= 1 for some 2 ≤ l ≤ n + 1. By induction hypothesis, the result follows immediately. P

Claim (ii): For any n1 , n2 ≥ 0, we have w = el [n1 ]ei [n2 ]ei1 [−1] · · · ein1 +n2 +1 [−1] = αj uj D tj sj , with each uj D tj sj ≤ w. We show Claim (ii) by induction on n1 . If n1 = 0, then X m w = (el [0]ei1 [−1] − αl,i e [−1])ei [n2 ]ei2 [−1] · · · ein1 +n2 +1 [−1] 1 m +

X

1≤m≤4

m αl,i e [−1]ei [n2 ]ei2 [−1] · · · ein1 +n2 +1 [−1]. 1 m

1≤m≤4

By Claim (i), the result follows immediately. If n1 > 0, then X m w = D n1 (el [0]ei1 [−1] − αl,i e [−1])ei [n2 ]ei2 [−1] · · · ein1 +n2 +1 [−1] 1 m −

X

1≤m≤4

Cnp1 el [p]ei1 [n1 − 1 − p]ei [n2 ]ei2 [−1] · · · ein1 +n2 +1 [−1]

0≤p≤n1 −1

+

X

m αl,i e [n1 − 1]ei [n2 ]ei2 [−1] · · · ein1 +n2 +1 [−1]. 1 m

1≤m≤4

By induction hypothesis, the result follows immediately. For any t ∈ N, u ∈ [D ω X], if wt((D t fij )u) = −1, |u| > 0 and (D t fij )u 6= ei [t](e1 [−1])t+1 , then by Claims (i) and (ii), we have X X m ei [p]ej [t−1−p]u+ αi,j em [t−1]u ≡ 0 mod(S, (D t fij )u). (D t fij )u−(D t fij )u = 0≤p≤t−1

1≤m≤4

Since for any t ∈ N, j 6= l, each critical common multiple of D t fij ∧ D t fil have form w = ei [t]ej [−1]el [−1]ei1 [−1] · · · eit−1 [−1], we get (D t fij , D t fil )w ≡ w − w ≡ 0 mod(S, w). 16

For the case of D t1 fi1 j ∧ D t2 fi2 j , where t1 6= t2 or i1 6= i2 , the proof is almost the same. So S is a Gröbner-Shirshov basis in Nov(X).

4.2

PBW type theorem in Shirshov form

Theorem 3. (PBW type theorem in Shirshov form) Let Nov(X) be a free Novikov algebra, k{X} be a free commutative differential algebra, S ⊆ Nov(X) and S c a Gröbner-Shirshov basis in k{X}, which is obtained from S by Buchberger-Shirshov algorithm. Then (i) S ′ = {uD ms | s ∈ S c , u ∈ [D ω X], m ∈ N, wt(uD ms) = −1} is a Gröbner-Shirshov basis in Nov(X). (ii) The set Irr(S ′ ) = {w ∈ [D ω X] | w 6= uD t s, u ∈ [D ω X], t ∈ N, s ∈ S c , wt(w) = −1} = Nov(X) ∩ Irr[S c ] is a linear basis of Nov(X|S). Thus, any Novikov algebra Nov(X|S) is embeddable into its universal enveloping commutative differential algebra k{X|S}. Proof. (i). We first show that any h ∈ S c has the form h = 0, wt(wi ) = wt(wi′ ), i, i′ ∈ Ih . Suppose f=

X

P

i∈Ih

γi wi , with each γi 6=

βi wi , with wt(wi ) = wt(wi′ ) for any i, i′ ∈ If ,

i∈If

g=

X

βi wi , with wt(wi ) = wt(wi′ ), for any i, i′ ∈ Ig ,

i∈Ig

and ′

(D t f, D t g)w′ =

X 1 1 ′ uD t f − vD t g = γj wj in k{X}. α1 α2 j∈J

Then it is obvious that wt(wj ) = wt(wj ′ ), ∀j, j ′ ∈ J. So whenever we add some non-trivial composition to S while doing the Buchberger-Shirshov algorithm, any monomial of such a composition will share the same weight. It follows that S ′ ⊆ Nov(X). If w = w1 D t1 (u1 D m1 s1 ) = w2 D t2 (u2 D m2 s2 ) ∈ Nov(X) is a non-trivial common multiple of D t1 (u1 D m1 s1 ) and D t2 (u2 D m2 s2 ), where s1 , s2 ∈ S c , t1 , t2 ∈ N, f = u1 D m1 s1 , g = u2 D m2 s2 ∈ S ′ , then by Theorem 1, we have (D t1 f, D t2 g)w =

X 1 1 δl ul D jl sl , w1 D t1 (u1 D m1 s1 ) − w2 D t2 (u2D m2 s2 ) = α1 α2 l∈L 17

where each δl ∈ k, ul ∈ [D ω X], sl ∈ S c , jl ∈ N, ul D tl sl < w. Furthermore, by Lemma 3.3, we can assume that for each l ∈ L, wt(ul D jl sl ) = −1, which means (D t1 f, D t2 g)w ≡ 0 mod(S ′ , w). So S ′ is a Gröbner-Shirshov basis in Nov(X). It remains to show that the ideal Id(S) of Nov(X) generated by S is Id(S ′ ). It is clear P βi ui D ti si , that S ⊆ Id(S ′). Since S c ⊆ Id[S c ] = Id[S], for any s ∈ S c , we have s =

where each βi ∈ k, ui ∈ [D ω X], si ∈ S and wt(ui D ti si ) = wt(s). By Lemma 3.2, it follows that S ′ ⊆ Id(S). (ii). Since {w ∈ [D ω X] | w 6= uD t s, u ∈ [D ω X], t ∈ N, s ∈ S ′ , wt(w) = −1} = {w ∈ [D ω X] | w 6= uD t s, u ∈ [D ω X], t ∈ N, s ∈ S c , wt(w) = −1} by (i), we have Irr(S ′) ⊆ Irr[S c ]. The result follows immediately. Remark 4.1. Theorem 3 essentially offers another way to calculate Gröbner-Shirshov basis in Nov(X) and it indicates some close connection between Nov(X|S) and its universal enveloping algebra k{X|S}. In fact, by Lemma 3.2, we have Nov(X) ∩ Id[S] = Id(S). Since Id[S] is a subalgebra of k{X} as differential algebra, it is a subalgebra of (k{X}, ◦) as Novikov algebra. Then we have a Novikov algebra isomorphism as follows: Nov(X)/Id(S) = Nov(X)/(Id[S] ∩ Nov(X)) ∼ = (Nov(X) + Id[S])/Id[S] ≤ (k{X|S}, ◦).

4.3

Word problems

In this subsection, we will show that the word problem of a Novikov algebra Nov(X|S) is solvable, if S consists of finitely many homogeneous relations. We also get a similar result for the commutative differential algebras with one derivative operator over a field of characteristic 0. Let k{X|S} be a commutative differential algebra and S = {fi | 1 ≤ i ≤ p}, p ∈ N, P where S is D ∪ X-homogeneous in the sense that for any f = qj=1 βj wj ∈ S, we have

|w1 |D∪X = |w2 |D∪X = · · · = |wq |D∪X . We will show that k{X|S} has a solvable word problem. In this subsection, we always assume that S ⊂ k{X} is a non-empty D∪X-homogeneous set. We call S a minimal set, if there are no f, g ∈ S with f 6= g, such that f = uD t g for any u ∈ [D ω X], t ∈ N. For any f and g ∈ S, if f = uD t g and the composition [f, D t g]f =

1 f α1

−

1 uD t g α2

≡ 0 mod(S, w), then we delete f from S to reduce the set S 18

in one step to a new set S0 , i.e., S −→ S0 = S \ {f }; If f = uD t g and the composition [f, D t g]f =

1 f α1

−

1 uD t g α2

6≡ 0 mod(S, w), then we replace f by h ,

1 f α1

−

1 uD t g α2

to

reduce the set S in one step to a new set S0 , i.e., S −→ S0 = (S \ {f }) ∪ {h}, where α1 = LC(f ) and α2 = LC(D t g). In both cases, we say that f is reduced by g. It is clear that S0 is also a D ∪ X-homogeneous set. Lemma 4.1. If |S| < ∞ and S is D ∪ X-homogeneous, then we can effectively reduce S into a minimal D ∪ X-homogeneous set S (0) in finitely many steps, such that Id[S] = P Id[S (0) ] and for any f ∈ S, we have f = βq uq D tq sq , with |uq D tq sq |D∪X = |f |D∪X and uq D tq sq ≤ f , where each βq ∈ k, uq ∈ [D ω X], sq ∈ S (0) , tq ∈ N.

Proof. Suppose S = {fi | 1 ≤ i ≤ p}, p ∈ N and 1 ≤ |f1 |D∪X ≤ |f2 |D∪X ≤ · · · ≤ |fp |D∪X . Given f, g ∈ S, suppose f = an [in ] · · · a1 [i1 ] and g = bm [jm ] · · · b1 [j1 ], with an [in ] ≥ · · · ≥ a1 [i1 ], bm [jm ] ≥ · · · ≥ b1 [j1 ] and jm ≤ in . To decide whether g can reduce f or not, we only need to check whether one of g, D 1 g, . . . , D in −jm g is a subword of f or not. Define ord(S) = (p, fp , fp−1, . . . , f1 ). Then if one reduce S in one step to S01 , we have ord(S01 ) < ord(S) lexicographically. Therefore, S can be reduced into a minimal set S (0) in finitely many steps, say, S −→ S01 −→ S02 −→ . . . −→ S0l = S (0) . Then by induction on l, we easily get each Id[S0m ] = Id[S] and for any f ∈ S, we have P f = βq uq D tq sq , with |uq D tq sq |D∪X = |f |D∪X and uq D tq sq ≤ f , where 1 ≤ m ≤ l, βq ∈ k, uq ∈ [D ω X], sq ∈ S0m , tq ∈ N.

Suppose that S is a minimal set and S = S (0) = {fi | 1 ≤ i ≤ p}, p ∈ N, where 1 ≤ |f1 |D∪X ≤ |f2 |D∪X ≤ · · · ≤ |fp |D∪X . For any f, g ∈ S (0) , t1 , t2 ∈ N, t1 , t2 ≤ 1, w = lcm(D t1 f , D t2 g), we will check composition [D t1 f, D t2 g]w whenever w is a nontrivial common multiple of D t1 f and D t2 g. If all such compositions are trivial, we just set S1 = S (0) . Otherwise, if for some t1 , t2 ≤ 1, w = lcm(D t1 f , D t2 g), the non-trivial P composition [D t1 f, D t2 g]w = i∈I βi wi = h, then |h|D∪X = |w|D∪X ≥ 2. We collect all such h to make a new set H0 and denote S1 = S (0) ∪ H0 . It is clear that each h ∈ H0 is 19

D ∪ X-homogeneous and we call |h|D∪X the D ∪ X-length of h. Now we reduce S1 to a minimal set S (1) . Noting that S (0) is a minimal set, if some inclusion composition is not trivial, then it must involve some element that is not in S (0) . Furthermore, each h ∈ H0 has D ∪ X-length at least 2, so every nontrivial inclusion composition that is added also (0)

(0)

has D ∪ X-length at least 2. So if we denote S (1) = Ssub ∪ R(0) , where Ssub = S (1) ∩ S (0) (0)

and R(0) = S (1) \ S (0) , then we get each r ∈ R(0) , |r|D∪X ≥ 2. For any f, g ∈ Ssub , if [D t1 f, D t2 g]w ≡ 0 mod(S1 , w), then [D t1 f, D t2 g]w ≡ 0 mod(S (1) , w) by Lemma 4.1. Continue this progress, and suppose (n−1)

Sn = S (n−1) ∪ Hn−1 , S (n) = Ssub

∪ R(n−1) ,

where S (n) is a minimal set and for any h ∈ Hn−1 , r ∈ R(n−1) , |h|D∪X ≥ n + 1, |r|D∪X ≥ n + 1. Then in order to get Sn+1 , for any f, g ∈ S (n) , t1 , t2 ∈ N, t1 , t2 ≤ n + 1, w = lcm(D t1 f , D t2 g), we need to check composition [D t1 f, D t2 g]w whenever w is non-trivial. If all such compositions are trivial, we just set Sn+1 = S (n) ; Otherwise, say [D t1 f, D t2 g]w = (n−1)

h is not trivial. If f, g ∈ Ssub

⊆ Sn , 0 ≤ t1 , t2 ≤ n, then by the construction,

[D t1 f, D t2 g]w ≡ 0 mod(Sn , w), and by Lemma 4.1, we get [D t1 f, D t2 g]w ≡ 0 mod(S (n) , w). Therefore, if we have a non-trivial composition, at least one of f and g is in R(n−1) , or at least one of t1 and t2 equals n + 1. Thus, if we denote Sn+1 = S (n) ∪ Hn , then for any h ∈ Hn , |h|D∪X ≥ n + 2. By the same reasoning as above, if we continue to reduce Sn+1 (n)

to a minimal set S (n+1) = Ssub ∪ R(n) , then we get for all r ∈ R(n) , |r|D∪X ≥ n + 2. As a result, we get the following lemma. Lemma 4.2. For any f ∈ S (n) , if |f |D∪X ≤ n, then f ∈ S (l) for any l ≥ n. Proof. Noting that after we get S (n) , any composition that may be added afterwards has D ∪ X-length more than n, but f can not be reduced by any element which has D ∪ X-length more than n or by element in S (n) \ {f }. Define Se = {f ∈

[

S (n) | f ∈ S (|f |D∪X ) }.

n≥0

20

Then by Lemma 4.2, we have Se = {f ∈

[

S (n) | f ∈ S (l) , f or any l ≥ |f |D∪X }.

n≥0

e and Se is a Gröbner-Shirshov basis in k{X}. Lemma 4.3. Id[S] = Id[S]

e ⊆ Id[S]. Proof. Since Id[S] = Id[S (0) ] = · · · = Id[S (n) ] for any n ≥ 0, we have Id[S] P βq uq D tq sq , On the other hand, for any f ∈ S, if |f |D∪X = n, then by Lemma 4.1, f = e Therefore, Id[S] = Id[S]. e For any where each sq ∈ S (n) and |sq |D∪X ≤ n, i.e., sq ∈ S. e t1 , t2 ∈ N, w = lcm(D t1 f , D t2 g), let l , |f |D∪X + |g|D∪X + t1 + t2 . If there exists f, g ∈ S,

composition [D t1 f, D t2 g]w , then [D t1 f, D t2 g]w ≡ 0 mod(Sl+1 , w) by construction. And by P Lemma 4.1, we have [D t1 f, D t2 g]w ≡ 0 mod(S (l+1) , w), i.e., [D t1 f, D t2 g]w = βi ui D ti si , e we get where each si ∈ S (l+1) and |si |D∪X ≤ |w|D∪X < l + 1. Thus by the definition of S, e w). [D t1 f, D t2 g]w ≡ 0 mod(S,

Theorem 4. If |S| < ∞ and S is D ∪ X-homogeneous, then k{X|S} has a solvable word problem. Proof. For any f =

P

βi wi ∈ k{X}, where w1 > w2 > . . . . We may assume that e implies that w1 = uD t s for some |wi |D∪X ≤ n. By Lemma 4.3 and Theorem 1, f ∈ Id[S] e then s ∈ S (n) . Note that S (n) is a finite e Moreover, if w1 = uD t s for some s ∈ S, s ∈ S. D ∪ X-homogeneous set that can be constructed effectively from S. After reducing f by such s, we get a new polynomial f′ = f −

X β1 t uD s = βi′′ wi′ , LC(uD t s)

with each |wi′ |D∪X ≤ n. Continue to reduce f ′ by elements in S (n) . If finally we reduce f ′ by S (n) to 0, then f ∈ Id[S]. Otherwise, f ∈ / Id[S]. In particular, if w1 6= uD t s for any e t ∈ N, and thus f ∈ / Id[S]. s ∈ S (n) , t ∈ N, then w1 6= uD t s for any s ∈ S, P Since for any f ∈ Nov(X), if f = 1≤i≤n βi wi is homogeneous in the sense that |w1 | = · · · = |wn |, then f is D ∪ X-homogeneous because |w|D∪X = 2|w| + wt(w) for any

w ∈ [D ω X]. Given Nov(X|S), if |S| < ∞ and S is homogeneous, then taking S as a subset of k{X}, S is D ∪ X-homogeneous. Thus we can get a Gröbner-Shirshov basis Se

in k{X}. Then by Theorem 3 and Theorem 4, we immediately get the following theorem. 21

Theorem 5. If |S| < ∞ and S ⊆ Nov(X) is homogeneous, then Nov(X|S) has a solvable word problem.

5

A subalgebra of N ov(a)

We construct a non-free subalgebra A of the free Novikov algebra Nov(a) over a field of characteristic 0, which implies that the variety of Novikov algebras is not Schreier. By Proposition 1 in [13], we immediately get the following lemma. Lemma 5.1. ([13]) The following statements hold: (i) The rank of a free Novikov algebra is uniquely determined, where the rank means the number of free generators. (ii) In a free Novikov algebra of rank n, any set of n generators is a set of free generators. (iii) A free Novikov algebra of rank n can’t be generated by less than n elements. In this subsection, we consider the free Novikov algebra Nov(a) generated by one element a. Theorem 6. Let A = ha ◦ a, (a ◦ a) ◦ a, ((a ◦ a) ◦ a) ◦ ai be the subalgebra of the free Novikov algebra Nov(a) generated by the set {a ◦ a, (a ◦ a) ◦ a, ((a ◦ a) ◦ a) ◦ a}. Then A is not free. Proof. Suppose that A is free. Then by Lemma 5.1 (iii), we get rank(A) ≤ 3. If rank(A) = 3, then by Lemma 5.1, a ◦ a, (a ◦ a) ◦ a, ((a ◦ a) ◦ a) ◦ a are free generators. However, (a ◦ a) ◦ (((a ◦ a) ◦ a) ◦ a) = ((a ◦ a) ◦ a) ◦ ((a ◦ a) ◦ a), which means that a ◦ a, (a ◦ a) ◦ a, ((a ◦ a) ◦ a) ◦ a are not free generators. If rank(A) = 1 and f = β1 (a ◦ a) + β2 (a ◦ a) ◦ a +

X

βi wi

is a free generator of A, where each wi has length at least 4, then a ◦ a = γ1 f +

X

γj f ◦ f ◦ f ◦ · · · ◦ f, 22

where f occurs at least twice in each term of the second summand on the right side and each of them is with some bracketing. We can rewrite this formula to the following form: a[0]a[−1] = γ1 f +

X

λi1 ,i2 ,...,in (D i1 f )(D i2 f ) · · · (D in f ),

where i1 ≥ i2 ≥ · · · ≥ in ≥ 0, n ≥ 2. Then each term in the second summand has leading term bigger than a[0]a[−1]. Since (D i1 f )(D i2 f ) · · · (D in f ) = (D i1 f )(D i2 f ) · · · (D in f ), by analysing the leading term of the left side and the right side, we get each λi1 ,i2 ,...,in = 0, so a[0]a[−1] = γ1 f, i.e., f =

1 a γ1

◦ a. However, (a ◦ a) ◦ a ∈ / ha ◦ ai = A. This is a

contradiction. If rank(A) = 2, suppose f1 = β1 (a ◦ a) + β2 ((a ◦ a) ◦ a) + f2 = γ1 (a ◦ a) + γ2 ((a ◦ a) ◦ a) +

X

X

βe we ,

γ e′ we′ ,

are free generators, where each we , we′ has length at least 4. Say a ◦ a = λ1 f1 + λ2 f2 +

X

λj1 ,j2 ,...,jn fj1 ◦ fj2 ◦ · · · ◦ fjn ,

X

µq1,q2 ,...,qm fq1 ◦ fq2 ◦ · · · ◦ fqm , X ((a ◦ a) ◦ a) ◦ a = ν1 f1 + ν2 f2 + νl1 ,l2 ,...,lr fl1 ◦ fl2 ◦ · · · ◦ flr , (a ◦ a) ◦ a = µ1 f1 + µ2 f2 +

where j1 , . . . , jn , q1 , . . . , qm , l1 , . . . , lr ∈ {1, 2}, and each term in the third summand on the right side of each equation is with some bracketing. Rewriting the right sides into linear combination of basis of the free Novikov algebra Nov(a) and comparing terms of length 2 and 3 on the left sides and the right sides, we get      β γ λ µ 1 0  1 1  1 1 =   β2 γ2 λ2 µ 2 0 1 and

 

β1 β2

    γ1 ν 0   1 =   . γ2 ν2 0 23

So ν1 = ν2 = 0 and ((a ◦ a) ◦ a) ◦ a =

X

νl1 ,l2 ,...,lr fl1 ◦ fl2 ◦ · · · ◦ flr .

However, among the terms of the right side, only (a ◦ a) ◦ (a ◦ a) has length 4, but ((a ◦ a) ◦ a) ◦ a 6= β(a ◦ a) ◦ (a ◦ a), for any β ∈ k. Therefore, A is not free.

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